Pawel Blasiak

The general boson normal ordering problem

Abstract We solve the boson normal ordering problem for F[(a † ) r a s ] , with r , s positive integers, [a,a † ]=1 , i.e., we provide exact and explicit expressions for its normal form N {F[(a † ) r a s ]} , where in N (F) all a s are to the right. The solution involves integer sequences of numbers which are generalizations of the conventional Bell and Stirling numbers whose values they assume for r = s =1. A comprehensive theory of such generalized combinatorial numbers is given including closed-form expressions (extended Dobinski-type formulas) and generating functions. These last are special expectation values in boson coherent states.

Combinatorial approach to generalized Bell and Stirling numbers and boson normal ordering problem

We consider the numbers arising in the problem of normal ordering of expressions in boson creation a† and annihilation a operators ([a,a†]=1). We treat a general form of a boson string (a†)rnasn…(a†)r2as2(a†)r1as1 which is shown to be associated with generalizations of Stirling and Bell numbers. The recurrence relations and closed-form expressions (Dobinski-type formulas) are obtained for these quantities by both algebraic and combinatorial methods. By extensive use of methods of combinatorial analysis we prove the equivalence of the aforementioned problem to the enumeration of special families of graphs. This link provides a combinatorial interpretation of the numbers arising in this normal ordering problem.

Boson normal ordering via substitutions and Sheffer-type polynomials

Abstract We solve the boson normal ordering problem for ( q ( a † ) a + v ( a † ) ) n with arbitrary functions q and v and integer n , where a and a † are boson annihilation and creation operators, satisfying [ a , a † ] = 1 . This leads to exponential operators generalizing the shift operator and we show that their action can be expressed in terms of substitutions. Our solution is naturally related through the coherent state representation to the exponential generating functions of Sheffer-type polynomials. This in turn opens a vast arena of combinatorial methodology which is applied to boson normal ordering and illustrated by a few examples.

ONE-PARAMETER GROUPS AND COMBINATORIAL PHYSICS

In this communication, we consider the normal ordering of sums of elements of the form (a*^r a a*^s), where a* and a are boson creation and annihilation operators. We discuss the integration of the associated one-parameter groups and their combinatorial by-products. In particular, we show how these groups can be realized as groups of substitutions with prefunctions.

Some useful combinatorial formulas for bosonic operators

We give a general expression for the normally ordered form of a function F[w(a,a†)] where w is a function of boson creation and annihilation operators satisfying [a,a†]=1. The expectation value of this expression in a coherent state becomes an exact generating function of Feynman-type graphs associated with the zero-dimensional quantum field theory defined by F(w). This enables one to enumerate explicitly the graphs of given order in the realm of combinatorially defined sequences. We give several examples of the use of this technique, including the applications to Kerr-type and superfluidity-type Hamiltonians.

Hierarchical Dobinski-type relations via substitution and the moment problem

We consider the transformation properties of integer sequences arising from the normal ordering of exponentiated boson ([a, a†] = 1) monomials of the form exp[λ(a†)ra], r = 1, 2, ..., under the composition of their exponential generating functions. They turn out to be of Sheffer type. We demonstrate that two key properties of these sequences remain preserved under substitutional composition: (a) the property of being the solution of the Stieltjes moment problem; and (b) the representation of these sequences through infinite series (Dobinski-type relations). We present a number of examples of such composition satisfying properties (a) and (b). We obtain new Dobinski-type formulae and solve the associated moment problem for several hierarchically defined combinatorial families of sequences.

Normal order: combinatorial graphs

A conventional context for supersymmetric problems arises when we consider systems containing both boson and fermion operators. In this note we consider the normal ordering problem for a string of such operators. In the general case, upon which we touch briefly, this problem leads to combinatorial numbers, the so-called Rook numbers. Since we assume that the two species, bosons and fermions, commute, we subsequently restrict ourselves to consideration of a single species, single-mode boson monomials. This problem leads to elegant generalisations of well-known combinatorial numbers, specifically Bell and Stirling numbers. We explicitly give the generating functions for some classes of these numbers. In this note we concentrate on the combinatorial graph approach, showing how some important classical results of graph theory lead to transparent representations of the combinatorial numbers associated with the boson normal ordering problem.

COMBINATORIAL PHYSICS, NORMAL ORDER AND MODEL FEYNMAN GRAPHS

The general normal ordering problem for boson strings is a combinatorial problem. In this talk we restrict ourselves to single-mode boson monomials. This problem leads to elegant generalisations of well-known combinatorial numbers, such as Bell and Stirling numbers. We explicitly give the generating functions for some classes of these numbers. Finally we show that a graphical representation of these combinatorial numbers leads to sets of model field theories, for which the graphs may be interpreted as Feynman diagrams corresponding to the bosons of the theory. The generating functions are the generators of the classes of Feynman diagrams.

Combinatorial Coherent States via Normal Ordering of Bosons

We construct and analyze a family of coherent states built on sequences of integers originating from the solution of the boson normal ordering problem. These sequences generalize the conventional combinatorial Bell numbers and are shown to be moments of positive functions. Consequently, the resulting coherent states automatically satisfy the resolution of unity condition. In addition they display such non-classical fluctuation properties as super-Poissonian statistics and squeezing.

Dobinski-type relations and the Log-normal distribution

We consider sequences of generalized Bell numbers B(n), n = 1, 2, ..., which can be represented by Dobinski-type summation formulae, i.e. B(n) = 1/C ∑k = 0∞ [P(k)]n/D(k), with P(k) a polynomial, D(k) a function of k and C = const. They include the standard Bell numbers (P(k) = k, D(k) = k!, C = e), their generalizations Br,r(n), r = 2, 3, ..., appearing in the normal ordering of powers of boson monomials (P(k) = (k+r)!/k!, D(k) = k!, C = e), variants of ordered Bell numbers Bo(p)(n) (P(k) = k, D(k) = (p+1/p)k, C = 1 + p, p = 1, 2 ...), etc. We demonstrate that for α, β, γ, t positive integers (α, t ≠ 0), [B(αn2 + βn + γ)]t is the nth moment of a positive function on (0, ∞) which is a weighted infinite sum of log-normal distributions.